_{1}

^{*}

In this paper, we consider an integral basis for affine vertex algebra
V_{k} (sl
_{2}) when the level
k is integral by a direct calculation, then use the similar way to analyze an integral basis for Virasoro vertex algebra
V
_{vir} (2
k,0). Finally, we take the combination of affine algebras and Virasoro Lie algebras into consideration. By analogy with the construction of Lie algebras over Z using Chevalley bases, we utilize the Z-basis of
L_{av} whose structure constants are integral to find an integral basis for the universal enveloping algebra of it.

While vertex algebras are usually assumed to be vector spaces over ℂ , the most important formula Jacobi identity makes sense over any commutative ring, so it is natural to consider vertex algebras over ℤ . An integral basis of a vertex algebra could be considered an analogue of the Chevalley basis in a Lie algebra. Similar to the construction of Lie algebras over ℤ using Chevalley bases, we can create vertex algebras over ℤ . Integral bases for vertex operator algebras associated with lattices have been studied in [

Next, we consider the Virasoro vertex algebra. Among the most important vertex algebras are those associated with the Virasoro Lie algebra. It has been studied in [

We know that affine Lie algebra and Virasoro Lie algebra have close relationship in physics, so we consider them simultaneously, i.e., as one algebraic structure. Then the definition of affine-Virasoro was introduced [

In this paper, we observe that the ℂ -basis of affine vertex algebra V k ( s l 2 ) and Virasoro vertex algebra V V i r ( 2 k , 0 ) may be integral basis for them in certain conditions. We create the conditions and confirm that they are exactly the integral bases. Then we utilize the analogue of Chevalley bases for finite dimensional Lie algebras to get an integral basis for the universal enveloping algebra of affine-Virasoro algebra.

We assume that the readers are familiar with the theory of vertex operator algebras [

Given an (untwisted) affine Lie algebra g ^ = g ⊗ ℂ [ t , t − 1 ] ⊕ ℂ k equipped with the bracket relation,

[ a ⊗ t m , b ⊗ t n ] = [ a , b ] ⊗ t m + n + m 〈 a , b 〉 δ m + n , 0 k

for a , b ∈ g and m , n ∈ ℤ , together with the condition that k is a nonzero central element of g ^ . Let k ∈ ℂ , g ^ ( − ) and g act trivially on ℂ and let k act as the scalar k, making ℂ a g ^ ( ≤ 0 ) -module, which we denote by

V g ^ ( k , 0 ) = U ( g ^ ) ⊗ U ( g ^ ( ≤ 0 ) ) ℂ k .

By the Poincaré-Birkhoff-Witt theorem, we have that,

V g ^ ( k , 0 ) = U ( g ^ ( + ) ) ≃ S ( g ^ ( + ) ) (2.1)

as a

Then

for

Now we have that,

where

In the next section, we will consider the integral basis of it when

In this section shall find an integral basis for

Definition 3.1. Suppose that

In order to find an integral basis for

where

is a

For convenience, we denote

Theorem 3.2. The formula (3.1) is an integral basis of

Proof. It is known that formula (3.1) is a

for

Since

where

In this section we shall find an integral basis for the Virasoro vertex algebra

Firstly we recall the definition of Virasoro vertex algebra [

The Virasoro algebra

together with the condition that

where

is a

We also have the graded subalgebras

Let

From the Poincaré-Birkhoff-Witt theorem, as a vector space,

Set

Then

where

for

and we have

where

Next, we will consider the integral basis of it when

is a basis of

with

for any

Just like the affine case, we have that,

where

and

the expression

is a

In this section we take the combination of affine algebras and Virasoro Lie algebras into consideration. By analogy with the construction of Lie algebras over

Firstly we recall the definition of the affine-Virasoro algebra [

Definition 5.1. Let L be a finite-dimensional Lie algebra with a non-degenerated invariant normalized symmetric bilinear form

with Lie bracket:

where

Now we consider the case of

where

Lemma 5.2. If

is an integral basis of

Proof. Since

Corollary 5.3. Let

where

Proof. We only need to check that

Since

we conclude that

In this paper, we get the integral basis for

The author declares no conflicts of interest regarding the publication of this paper.

Wang, A. (2020) Integral Basis of Affine Vertex Algebra and Virasoro Vertex Algebra . Journal of Applied Mathematics and Physics, 8, 652-659. https://doi.org/10.4236/jamp.2020.84050